Theorem toponcom 22871

Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
toponcom	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Proof of Theorem toponcom

Step	Hyp	Ref	Expression
1		toponuni 22857	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐽 = ∪ 𝐾)
2	1	eqcomd 2742	. . 3 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐾 = ∪ 𝐽)
3	2	anim2i 617	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
4		istopon 22855	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
5	3, 4	sylibr 234	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∪ cuni 4888  ‘cfv 6536  Topctop 22836  TopOnctopon 22853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-topon 22854

This theorem is referenced by:  toponcomb  22872  kgencn3  23501